Fully copositive matrices

Abstract

The class of fully copositive (C₀^f) matrices introduced in [G.S.R. Murthy, T. Parthasarathy, SIAM Journal on Matrix Analysis and Applications 16 (4) (1995) 1268-1286] is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class of Q₀-matrices are P₀-matrices. As a corollary of this main result, we establish that a bisymmetric Q₀-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization of Q₀-matrices within the class of C₀^f. While establishing this characterization, it will be shown that Graves's principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable to C₀^f ∩ Q₀ class. As a byproduct of this characterization, we observe that a C₀^f-matrix is in Q₀ if, and only if, it is completely Q₀. Also, from Aganagic and Cottle's [M. Aganagic, R. W. Cottle, Mathematical Programming 37 (1987) 223-231] result, it is observed that LCPs arising from C₀^f ∩ Q₀ class can be processed by Lemke's algorithm.